The authors consider two natural approaches to the boundary at infinity of hyperbolic metric spaces. The Gromov boundary is generated as a set of equivalence classes of sequences converging to infinity in Gromov’s sense. Converging to infinity sequences \(x_n\) and \(y_n\) are equivalent if Gromov product \((x_n\cdot y_n)\) has infinite limit when \(n\to\infty\). The metric compactification \(\overline X^d\) of a locally compact metric space \((X,d)\) corresponds to the pure states of the commutative unital \(C^*\)-algebra generated by constants, functions that vanish at infinity and differences of distant functions \(d_x(y)=d(x,y)\). The metric boundary is by definition \(\partial_dX=\overline X^d\setminus X\).
The main result of the paper is
Theorem. Let \((X,d)\) be a proper, \(0\)-hyperbolic metric space with a countable base. Then its Gromov boundary and metric boundary are homeomorphic.
In general case of \(\delta\)-hyperbolic space the authors show that the Gromov boundary is a quotient of the metric boundary with continuous quotient map. As a corollary, the action of any word-hyperbolic group \(G\) on its Cayley graph is amenable. Another application of the result is the fact that the metric boundary of the complete proper \(0\)-hyperbolic metric space has no non-Busemann points.